{"paper":{"title":"Smirnov Decomposition of a Horizontal Vector Charge in the Heisenberg Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Divergence-free horizontal currents in the Heisenberg group decompose into a measure on horizontal curves via a direct application of the horizontal Liouville theorem.","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Wilhelm Klingenberg, Zhengyao Huang","submitted_at":"2026-05-12T20:23:27Z","abstract_excerpt":"A divergence-free horizontal vector current in Heisenberg space may be viewed as an element of the dual space of horizontal test vector fields. By applying a horizontal Liouville theorem in this setting, the flow lines of such a vector field generate a family of horizontal curves and an associated measure on this collection. In this paper, we provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The horizontal Liouville theorem applies in this setting to generate the family of horizontal curves and the associated measure from the divergence-free current.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A direct proof shows that a divergence-free horizontal vector current in the Heisenberg group decomposes as a measure on horizontal curves via the horizontal Liouville theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Divergence-free horizontal currents in the Heisenberg group decompose into a measure on horizontal curves via a direct application of the horizontal Liouville theorem.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0f3a9539b5b5033ed34d67d81b3debf1dd8c306465fd48915fe4c95c322d851b"},"source":{"id":"2605.12716","kind":"arxiv","version":1},"verdict":{"id":"bbaaa10c-78d8-45ed-9544-4f000da611b8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T21:37:48.074076Z","strongest_claim":"we provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution.","one_line_summary":"A direct proof shows that a divergence-free horizontal vector current in the Heisenberg group decomposes as a measure on horizontal curves via the horizontal Liouville theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The horizontal Liouville theorem applies in this setting to generate the family of horizontal curves and the associated measure from the divergence-free current.","pith_extraction_headline":"Divergence-free horizontal currents in the Heisenberg group decompose into a measure on horizontal curves via a direct application of the horizontal Liouville theorem."},"references":{"count":15,"sample":[{"doi":"","year":null,"title":"Analysis and Geometry in Metric Spaces , doi =","work_id":"a7308f10-b3f1-4f01-a943-773c77bcb597","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Marco Carfagnini , journal=. 2021 , url=","work_id":"55241214-dfeb-4ed8-9de6-a56d4af563f0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s12220-010-9205-5","year":2012,"title":"2012 , pages =","work_id":"99dad295-b00e-4322-aae9-fa52422183cb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"A comprehensive introduction to sub-Riemannian geometry , publisher=","work_id":"90b6f8ad-690f-47ad-9e44-7c71256828e9","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Valentino Magnani and Dario Trevisan , journal=. 2016 , url=","work_id":"36975c74-f754-4053-a809-0d66848f0d81","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"f6ff2fa1820715ec1f1512c1bd7f61b85ee13da1664a4d60345e2c907e5ddb5a","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9a42ff593316b571c72f133840a0c4670f06a2d29fd8ff0013390c83621a2390"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}